The new Xbox One has yet to implement compatibility with third-party headsets, meaning for now, you're stuck using one of the official Xbox One headsets available. However, many have run into problems with the included wired headsets, and would much rather use the expensive third-party ones they already have for the Xbox 360.
Since the Xbox One controllers use a USB adapter, the Xbox 360 headsets you have, including ones by Turtle Bay and Tritton, simply won't work. But instead of throwing them out or letting the dust accrue, you can actually splice the wires of your favorite Xbox 360 headset to the USB connector of new junk Xbox One headset to make it work.
You will need a wire cutter and stripper, the Xbox One headset, an Xbox 360 headset, and electrical tape. Basically, you just cut and strip the wires of the Xbox 360 headset near the jack, and cut the Xbox One headset's jack off and strip the wires on that, then splice the two together. The most important part is making sure to connect the correct wires to its correct counterpart.
If you have an older third-party Xbox 360 headset, this is how the wires should be paired.
If you have newer Xbox 360 cables, this may be an alternative.
Once you have spliced the correct wires together and made sure the contacts are good, use the electrical tape to cover it up. For detailed instructions on doing this, check out the complete guide by Instructables user DragonballZ Deep.
For Tritton brand headsets, the wire breakdown may look like this:
Take a look at the video by Mike Ridolfi to see all of the steps on converting your Xbox 360 Tritton headset to work on the Xbox One.
NOTE: Using any of these methods will void your headsets' warranty and could damage both of them. Proceed at your own risk.